(1) Let \(p\)  be a minimal (undirected) path between \((M_1,\xi _1)\)  and \((M_2,\xi _2)\)  in \(\Gamma \) . If every vertex in \(p\)  corresponds to an overtwisted contact manifold the distances in \(\Gamma \)  and \(\Gamma _{\textrm {OT}}\)  agree. In general, however, the path \(p\)  might run through tight contact manifolds. To prevent this we choose an ordered surgery link \(L\)  in \((M_1,\xi _1)\)  such that contact \((\pm 1)\) -surgery in that given order along \(L\)  corresponds to the path \(p\) . We denote by \(\xi _0\)  the overtwisted contact structure on \(S^3\)  with vanishing normalized \(\operatorname{d}_3\) -invariant. A 2-component surgery diagram of \((S^3,\xi _0)\)  is shown in Figure REF  (center). We add this surgery diagram in a Darboux ball in the exterior of \(L\)  to the surgery link, where we first perform the surgeries along the two new Legendrian knots and afterwards the surgeries corresponding to \(p\) . Then any contact manifold in the path \(p\)  is replaced by a connected sum with \((S^3,\xi _0)\) .
By Eliashberg's classification of overtwisted contact structures [1]}, it follows that we have constructed a new path \(p^{\prime }\)  from \((M_1,\xi _1)\)  to \((M_2,\xi _2)\)  in \(\Gamma _{\textrm {OT}}\)  of length 2 larger than the length of \(p\) .
(2) By Wand's theorem [2]}, any directed path in \(\Gamma \)  between two overtwisted contact manifolds cannot run through a tight contact manifold, see proof of Theorem REF .
<FIGURE>